QUANTUM HALL EFFECTS
By
ZHENYUE ZHU
Bachelor of Science in Physics
Nanjing University
Nanjing, P. R. China
2002
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial ful¯llment of
the requirements for
the Degree of
MASTER OF SCIENCE
July, 2007
COPYRIGHT
By
Zhenyue Zhu
July, 2007
QUANTUM HALL EFFECTS
Thesis Approved:
Xincheng Xie
Thesis Advisor
John Mintmire
Paul Westhaus
K. S. Babu
A. Gordon Emslie
Dean of the Graduate College
ii
ACKNOWLEDGMENTS
First I would like to express my sincere appreciation to my thesis advisor Dr.
Xincheng Xie for his intelligent supervision and constructive guidance in my academic
research. Especially, I would like to thank him for his ¯nancial support through my
research assistantship so that I can fully devote myself in research for two years.
I also like to extend my thanks to other committee members: Dr. John Mint-
mire, Dr. Paul Westhaus, Dr. K. S. Babu. Their classes have greatly bene¯t my
physics education.
Moreover, I would like to express my sincere gratitude to those who have pro-
vided suggestions and assistance for my study: Dr. Ye Xiong, Dr. Qing-feng Sun, Dr.
Bin Chen, Dr. Zhong-Shui Ma, Dr. Clint Conner, Kai Wang, Yanzhi Meng, Junwen
Li and Feng Gao. These friends of mine have made graduate school a much more
enjoyable time.
To the sta® members in the physics department, I want to thank Susan Cantrell,
Cindi Raymond and Stephanie Hall for your help in all the matters outside physics.
Thank you for making sure I got paid every month so that I can live in a cozy life.
Finally, I thank my parents for their constant support and encouragement in
my whole school life. Their understanding and support have strengthened my belief
to follow my academic road.
iii
TABLE OF CONTENTS
Chapter Page
1. INTRODUCTION ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 1
1.1. Background ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 1
1.2. Classical Explanation :::::::::::::::::::::::::::::::::::::::::::::::: 2
2. Integer Quantum Hall E®ects::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5
2.1. Landau Levels :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5
2.2. The role of disorder ::::::::::::::::::::::::::::::::::::::::::::::::::: 6
2.3. Laughlin Gauge invariant argument :::::::::::::::::::::::::::::: 8
2.4. Current Carrying Edge States ::::::::::::::::::::::::::::::::::::: 10
3. Quantum Hall E®ect in a Four Terminal Device With Rashba Spin-
Orbital Interaction ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 13
3.1. Introduction to Rashba Spin Orbital Interaction:::::::::::::: 13
3.2. Tight binding model for a 2DES with Rashba SOI and
a perpendicular magnetic ¯eld:::::::::::::::::::::::::::::::::::: 15
3.3. Landauer-BÄuttiker Formulae for a Four Terminal De-
vice :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 17
3.4. Self Energy Term for a Normal Semi-in¯nite Lead ::::::::::: 20
3.5. Our numerical results about transport and localization
properties :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25
4. Fractional Quantum Hall E®ect :::::::::::::::::::::::::::::::::::::::::::::::::::: 31
4.1. The Laughlin wave function for a º = 1
2n+1 state ::::::::::::: 31
4.2. Properties of Laughlin Wave Function ::::::::::::::::::::::::::: 35
4.3. Quasi-hole Excitation and Statistics:::::::::::::::::::::::::::::: 36
4.4. Outlook of Fractional Quantum Hall E®ect :::::::::::::::::::: 38
BIBLIOGRAPHY::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 40
iv
LIST OF FIGURES
Figure Page
1.1. The experimental data of Hall conductance Rxy and magnetoresis-
tance Rxx of a 2D electronic system as a function of magnetic ¯eld. :::: 2
1.2. The sketch of the formation of the Hall conductance. The negative
charged electrons will accumulated on one side of the slab. The
transverse voltage drop is VH. ::::::::::::::::::::::::::::::::::::::::::::::::::::: 3
2.1. Density of states versus Fermi energy E for (a) without disorder
(delta function) (b) with disorder (Lorentz shape function). Ex-
tended states and localized states are indicated in (b). (c) The
Hall conductance versus Fermi energy E. When Fermi energy ¯lled
extended states in one LL, Hall conductance increased by e2=h.:::::::::: 7
2.2. Thought experiment of adiabatic increasing of magnetic quanta
through the loop. Current running along the loop. Magnetic °ux
© pass through the loop. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 8
2.3. The illustration of the movement of wave function centers as increas-
ing magnetic °ux with one quanta. ::::::::::::::::::::::::::::::::::::::::::::::: 9
2.4. The formation of the edge current is due to the truncate of the elec-
tron cyclotron orbits. Edge currents °ow in opposite directions at
di®erent edge. The bulk current is zero. :::::::::::::::::::::::::::::::::::::::: 10
2.5. The energy spectrum versus transverse direction y for a realistic con-
¯ned potential. Inside the bulk sample, eigenvalues remain un-
changed. But near the boundary, the rising of the eigenvalue is due
to the in¯nity potential at the boundary. ::::::::::::::::::::::::::::::::::::::: 11
3.1. Landau Levels of an electron as functions of ´ for g = 0:1 is the
Zeeman splitting energy. The cross of the LLs are indicated with
arrows in the graph. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 14
3.2. The sketch for 2D lattice model with i stands for row indices and j
for column indices. i and j may also labeled as in y and x axis. The
path for an electron to jump one unit box is indicated. ::::::::::::::::::::: 15
v
Figure Page
3.3. Sketch illustration of our numerical calculation model with four semi-
in¯nite leads. The Rashba SOI, disorder potential and magnetic
¯eld only exists in the central unshadowed region. ::::::::::::::::::::::::::: 18
3.4. Illustration to calculate the self energy for a semi-in¯nite lead. The
solid line connecting system is considered as Hamiltonian H0, the
dashed line connecting part is interaction Hamiltonian HI . :::::::::::::::: 22
3.5. A conductor connected with a normal lead p on the left side. A point
in lead p is labeled Pi if it is adjacent to point i in the conductor.
The dashed line is the connection of lead with conductor. The
hopping phases in the junction are labeled. The total phase for a
unit box in the junction is zero. So there is zero magnetic ¯eld in
the junction. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 24
3.6. The normalized localization length L/M vs disorder strength for a
long bar of length 100000 and width M=12, 24, 40. The parameters
are setup as E=-2.0 and ¸ = 0:5.:::::::::::::::::::::::::::::::::::::::::::::::::: 25
3.7. The normalized localization length versus Fermi energy E with
strength of disorder w = 0:5, Zeeman splitting energy g = 0:6,
B = e¼=4~ and system width M=40 at di®erent Rashba strength. :::::: 26
3.8. Fermi energy versus normalized localization length at di®erent
strength of disorder with ¸=0.3, zeeman splitting energy g = 0:6
and system width M=40. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 28
3.9. Hall conductance versus Fermi energy (dashed line) and normalized
localization length versus Fermi energy (solid line) with strength of
disorder w = 0:1 and Zeeman splitting energy g = 0:4. The size of
the system is 40 £ 40 and lead width is 20. :::::::::::::::::::::::::::::::::::: 29
3.10. Magnetic ¯eld versus Hall conductance with strength of disorder w =
0:1 and the Fermi energy E = 2:0. The zeeman splitting energy is
g = 1:2
¼ B. The size of the system is 40 £ 40 and lead width is 20.:::::::: 29
4.1. The density of eigen wave function ª0;n(z; ¹z) is a ring shape. :::::::::::::: 33
4.2. Exchange statistics for Fermion, boson and anyon. :::::::::::::::::::::::::::: 38
vi
CHAPTER 1
INTRODUCTION
In this chapter, we will review the experiments about quantum Hall e®ect and
apply classical electro-magnetic theory to explain the linear increasing of Hall re-
sistance with magnetic ¯eld. In chapter 2, we will brie°y explain the formation of
integer quantum Hall e®ect. In chapter 3, we will provide our numerical results about
localization and Hall e®ect in a disordered 2D electronic systems with Rashba spin-
orbital interaction. In the ¯nal chapter, we will introduce some basic exotic properties
of fractional quantum Hall e®ect and bring the outlook of fractional quantum Hall
e®ect.
1.1 Background
Even though the Hall e®ect was ¯rst observed by E. H. Hall in 1879,1 the integer
quantum Hall e®ect (IQHE)was ¯rst discovered by von Klitzing, Dorda and Pepper
in 1980. 2
The di±culty of studying QHE is due to the experimental realization of almost
ideal two dimensional electronic systems. In 2D electronic system, the electrons are
con¯ned to move only in two spatial dimensions. In the third dimension, they have
quantized energy levels. The high density 2D electron systems are typically created
in metal-oxide-semiconductor ¯eld e®ect transistor (MOSFET) and in semiconductor
heterojunctions.
Hall resistance exhibiting plateau structures in units of e2=h was observed in
2D electronic system at low temperature and strong magnetic ¯eld. The quantization
is so accurate that it is better than one part in ten million. Nowadays these accurate
plateaus will become a new standard for the de¯nition of resistance.
1
2
Figure 1.1. The experimental data of Hall conductance Rxy and magnetoresistance
Rxx of a 2D electronic system as a function of magnetic ¯eld.
In 1982, Tsui, Stormer and Gossard discovered fractional quantum Hall e®ect
(FQHE) in higher mobility samples, where Hall conductance exhibit plateaus to the
fractional multiples of e2=h.3 Figure 1.1 is taken from Stormer et al..4 From the graph,
we can see that the Hall resistance increases linearly with magnetic ¯eld at weak mag-
netic ¯eld. But with further increasing of magnetic ¯eld, Hall resistance will exhibit
integer or fractional plateaus, accompanied with minimum in the diagonal resistivity,
indicating a dissipationless current. Although similar experimental results exist be-
tween IQHE and FQHE, the physics reasons behind them is quite di®erent. In IQHE,
disorder plays an important role, while electron-electron interaction is predominant
in FQHE. In the next section, we will try to start from classical theory to partially
explain these phenomena.
1.2 Classical Explanation
Hall (conductivity) measurement in a weak magnetic ¯eld is a basic tool to
characterize a semiconductor ¯lm, because we can deduce the carrier density and
3
mobility. In this section, we will explain the Hall resistance linear dependent on
magnetic ¯eld at classical regime.
Figure 1.2. The sketch of the formation of the Hall conductance. The negative charged
electrons will accumulated on one side of the slab. The transverse
voltage drop is VH.
Consider a two dimensional conducting metal slab in x-y plane [Fig. 1.2], a
current I is °owing in the x direction. A magnetic ¯eld B is applied perpendicular to
the x-y plane in z direction. When electrons are moving in the slab, it feels the Lorentz
force which comes from the B ¯eld. So it will de°ect its motion and accumulate on
the other side of the slab. With further accumulation of electrons on one side of the
slab, the transverse electric ¯eld Ey will be established to cancel the Lorentz force. In
a steady state, electrons will only transport along the -x direction. The Hall voltage
is de¯ned as VH = EyW (W is the width of the slab). The total current along the x
direction is Ix = WJx. The Hall resistance is thus de¯ned as RH = VH=I = Ey=Jx.
The current density
¡!j
related to
¡!E
is given by
¡!j
= ¾
¡!E
, with ¾ is a conducting
matrix
¾ =
0
@¾xx ¾xy
¾yx ¾yy
1
A (1.1)
4
For simplicity, we will omit the electron scattering with impurities. In a steady state,
the electron motion satis¯es md¡!v
dt = ¡e(
¡!E
+ ¡!v
£
¡!B
) = 0 and jy = 0. From the
above equations, we can derive that
¾ =
0
@ 0 ne=B
¡ne=B 0
1
A; and ½ = ¾¡1 =
0
@ 0 ¡B=ne
B=ne 0
1
A (1.2)
The Hall resistivity is de¯ned as ½H = ¡½xy = B=ne with n is the electron
density for the system. From the above equation, we can see that the Hall resistivity
increase linearly with B ¯eld. This only explains the Hall resistance linearly depen-
dence with B ¯eld at weak B ¯eld, which corresponds to the classical Hall e®ect. For
the plateau structures, we will refer to quantum mechanics approach to explain these
phenomena.
CHAPTER 2
Integer Quantum Hall E®ects
In this chapter, we will provide several aspects: Landau levels, impurity e®etcs,
laughlin gauge invariant argument and edge state to understand the experimental ob-
served integer quantum Hall e®ects (IQHE). The integer quantized Hall conductance
can be understood as the ¯lling of Landau levels. If Fermi energy lies in the gap
between n and n+1 landau levels. The Hall conductance is ne2=h.
2.1 Landau Levels
The ¯lling of Laudau levels (LLs) is the key to understand the origin of IQHE.
First, let us solve the SchrÄodinger equation for an electron moving in a x-y plane with
a perpendicular magnetic ¯eld in z direction. The size of the system is L times W.
The vector potential
¡!A
is chosen as (-By, 0, 0). If we suppose that the electron is
polarized, we don't need to consider its spin degrees of freedom. The isotopic e®ective
mass Hamiltonian is given by:
H =
1
2m
(¡!p+ e
¡!A
)2 =
1
2m
[(px ¡ eBy)2 + p2y
] (2.1)
Where ¡e is the electron charge, m is the e®ective mass for the system. Separation
of variables x, y, we get the following wave function.
Ãkn = eikxÁn(y) (2.2)
With energy eigenvalues are En = ²ny +²x with ²x = ~2k2=2m. And Án(y) satisfy the
following harmonic oscillator type equation
·
p2y
2m
+
e2B2
2m
(y ¡ ~k
eB
)2 ¡ ~2k2
2m
¸
Án(y) = ²nyÁn(y) (2.3)
5
6
The harmonic wave function Án(y) centered at y = ~k=eB. k is the wave vector along
x direction. Using the quantization condition, we get k = 2¼j=L(j is an integer). And
²ny = (n+1=2)~! ¡~2k2=2m with cyclotron frequency ! = eB=m. The total energy
eigenvalue is
En = (n + 1=2)~! (2.4)
The wave function for the system can be understood as plane wave along x direction
and various harmonic oscillator located at di®erent y coordinates. The eigenvalues of
such 2DES are called Landau levels(LLs). n=0 is the ¯rst LL.
The number of states contained per LL is given by how many states can be
located along the y direction. The maximum number of states
N = W=¢y = WLeB=h = AeB=h (2.5)
with A = WL is the area of the 2DES. The electron density per LL is ne = N=A =
eB=h. Therefore if the chemical potential is between the n and n+1 LLs. n LLs will
be ¯lled. The total electron density is neB=h. From Eq. 1.2, we get ¾xy = ne2=h.
If integer number of LLs are ¯lled, the Hall conductance is a quantized value. From
Kubo linear response formulae, ¾xx only depends on the states at the Fermi surface.
If the Fermi energy lies in the gap between two close LLs, ¾xx = 0 at T = 0. At
T 6= 0, the diagonal conductance is exponentially small.
2.2 The role of disorder
Although the argument seems right, it does not apply to a dirty system. Be-
cause the electrons motion will interact with the random potential, this potential will
break the system's translational invariance. What's more, some states may be local-
ized by this impurity potential. Therefore localized states can't carry any current.
Prange and Ando ¯rst analyze electrons in the lowest LL interacting with a single ±
function impurity.5;6 They found that the conductance will remains the same quan-
tized value e2=h, because the current carried by the extended states will compensate
the non-current-carrying localized states. Although his calculation gives the robust-
ness of the quantization, it can't be applied to a realistic system where impurities are
7
randomly distributed. The LL will be broadened as a Lorentz shape centered at each
unperturbed LL with the in°uence of disorder.
Figure 2.1. Density of states versus Fermi energy E for (a) without disorder (delta
function) (b) with disorder (Lorentz shape function). Extended states
and localized states are indicated in (b). (c) The Hall conductance
versus Fermi energy E. When Fermi energy ¯lled extended states in
one LL, Hall conductance increased by e2=h.
As illustrated in Fig. [2.1], the states which are close to the center of LL are less
localized than at the edge of Lorentz distribution. Therefore from this picture, the
extended states only form at the center of LLs, all the other states are localized.7{10
As we ¯x the chemical potential (equivalent to Fermi energy at T = 0) and increase
the B ¯eld, gaps of di®erent LL ~! will increase also. When chemical potential is
tuned through these extended state, because only extended states will contribute
to the current, the current will jump discontinuously with e2=h. And the current
remains constant, if Fermi energy lies in the localized state region. So the plateau
structure comes from the range of magnetic ¯eld, where the number of extended
states are ¯xed. The only question left to answer is what if Fermi energy lies inside
the extended states, where one LL is partially ¯lled. Actually, a partially ¯lled LL
will lead to the formation of fractional Quantum Hall e®ect (FQHE). This is the
case where experiments show surprising results. We will brie°y introduce FQHE in
chapter 4.
8
2.3 Laughlin Gauge invariant argument
In this section, we will introduce a beautiful argument from Laughlin to explain
IQHE.11 The model is still a 2DES as in Fig. [2.2], but its two ends are bent into a
loop of length L with the x coordinates along the loop. Magnetic ¯eld B pieces the
loop everywhere and is normal to the surface. A Hall voltage drop ¢V is across the
edge of loop. The steady current is °owing the loop. Consider another magnetic °ux
© passing through the loop.
Figure 2.2. Thought experiment of adiabatic increasing of magnetic quanta through
the loop. Current running along the loop. Magnetic °ux © pass
through the loop.
The current I °owing in the loop can expressed as: 12;13
Ix =
@E
@©
=
@E
L@A
(2.6)
where E is the total electron energy, and A is the vector potential pointing around
the loop. If we adiabatically increase the °ux by ¢©, the vector potential will change
from A to A+¢A = A+¢©=L. So the electrons wave function will acquire a phase
as:
©(x) ! ©(x) exp(i
e¢©
~L
x) = ©(x) exp(2¼i
¢©
L©0
x) (2.7)
with magnetic °ux quanta ©0 = h=e. If the electrons are localized, the wave function
vanishes outside a localized region, which is much smaller than L. It will not respond
9
to the increasing °ux. So for the localized states, the energy change and current
is zero with adiabatically increasing of °ux. But for the extended states, such a
transformation is only allowed if ¢© =©0 is an integer. (Wave function must be a
single value at certain position, ©(x) = ©(x+L)). So ¢© = nh=e and ¢A = nh=eL.
Now let's look at if we add one °ux quanta into the system, how the electron
wave function will be changed. Initially, we could imagine that all the electrons wave
function are located at the centers
yi =
~kj
eB
=
~2¼
eBL
j =
h
eBL
j (2.8)
with j is an integer from 1 to N (maximum number of states per LL). The interval
of each center is ¢y = h=eBL. ¢y can expressed as ¢A=B. This relation exactly
means that if the magnetic °ux increase by one quanta, the centers of electron's wave
function at j will become j+1. If we consider the electron wave function as a whole,
Figure 2.3. The illustration of the movement of wave function centers as increasing
magnetic °ux with one quanta.
the increasing of one °ux quanta is equivalent to one electron moves from one edge of
loop to another. If n LLs are ¯lled, n electrons will move from one edge of the loop to
another. So the change of electrons energy ¢E = neVy with ¢© = h=e. The current
is then
Ix =
¢E
¢©
=
neVy
h
e
= n
e2
h
Vy (2.9)
10
The Hall conductance is thus
¾xy =
Ix
Vy
= n
e2
h
(2.10)
2.4 Current Carrying Edge States
Halperin reformulated Laughlin's argument in a annular geometry.14 He also
proposed that quasi-one dimensional states at the perimeter of the sample carrying
a current. And they don't localized in the presence of moderate strength disorder
potential. If Fermi levels at two edges of the sample are di®erent, the edge states will
contribute to the Hall conductance.
x
y
0
B
IR
IL
Figure 2.4. The formation of the edge current is due to the truncate of the electron
cyclotron orbits. Edge currents °ow in opposite directions at di®erent
edge. The bulk current is zero.
We can refer to a naive picture of electron cyclotron motion to understand
the non-vanishing edge current.15 Classically, electrons will move in circular orbits in
a magnetic ¯eld as the result of Lorentz force. In a bulk system, two neighboring
cyclotron orbits overlap, which will not contribute to the current. But this is not
the case at the edge of the sample. The cyclotron orbits are truncated at the two
edges can be regarded as a quasi-one dimensional current °owing in the opposite
directions. (Fig. [2.4]) The chirality of the edge current °ow is strong against disorder
potential. Because disorder potential will cause the backscattering of electrons, but
11
no backscattering exist in the chiral edge states. So the current is only determined
by the edge currents.
Now, we will start from the quantum mechanics to investigate the edge states.16
The model is the same as discussed in section 2.1. But we consider a more realistic
circumstance with a con¯ned potential with
V (y) =
8><
>:
0; y 2 [¡L=2; L=2]
1; otherwise
(2.11)
The potential means that near the boundaries, the potential increases smoothly to
in¯nity. The eigenvalues of the such as system will be modi¯ed as En = (n+1=2)~!+
V (y). Inside the sample, con¯ned potential is zero, the energy eigenvalues are the
same as before. But near the boundary of the sample, the eigenvalues will be quite
di®erent. The graph of En as a function of y is depicted in Fig. [2.5].
0
E
n=0
n=1
n=2
-L/2 L/2 y
Figure 2.5. The energy spectrum versus transverse direction y for a realistic con¯ned
potential. Inside the bulk sample, eigenvalues remain unchanged. But
near the boundary, the rising of the eigenvalue is due to the in¯nity
potential at the boundary.
The time dependent wave function far away from the boundary can be expressed
as:
ª(x; y; t) = exp(ikx ¡ i!kt)©n(y ¡ kl2B
) (2.12)
12
with ~!k = En, the magnetic length lB =
p
~=eB. ©n is a harmonic oscillator
function centered at y = kl2B
. At the edge of the sample y = L=2, the electrons
velocity is
vnk =
@Enk
~@k
=
@Enk
~@y
@y
@k
(2.13)
If we linearize the spectrum at two boundaries, @Enk
@y is a constant. However at the
two edges, @Enk
@y has di®erent signs. At boundary y = L=2, vnk > 0, at y = ¡L=2,
vnk < 0. So the current °owing at the two edges are in opposite direction. In the
center, the bulk LLs are constant as y. Hence @Enk
@y = 0 and vnk = 0. No current is
°owing is °owing away from the edge. The edge current is along the x direction, so
the edge states are actually quasi-one dimensional states. If n LLs are ¯lled, at each
boundary there will be n edge states. Now let's consider if the chemical potential at
the upper boundary (current moving in right direction) ¹R is slightly higher than the
lower boundary (current moving in left direction) ¹L. The higher local Fermi energy
will lead to higher electron density. For N edge states, the current moving in the right
direction is 17;18
IR = Nevnk
dn
dE
¹R (2.14)
For one dimensional case, the density of states is dn=dk = 1=2¼.
dn
dE
=
dn
dk
dk
dE
=
1
2¼~vnk
=
1
hvnk
(2.15)
So IR = Ne¹R=h and IL = Ne¹L=h. The total current °owing in the x direction is
Ix = IR ¡ IL =
Ne
h
(¹R ¡ ¹L) =
Ne
h
eVy =
Ne2
h
Vy (2.16)
Therefore, ¾xy = Ne2=h. If two voltage probes are located as the same side of the
edge, the voltage drop is zero (VL = 0), because at the same edge, the potential is
the same. ¾xx = 0.
CHAPTER 3
Quantum Hall E®ect in a Four Terminal Device
With Rashba Spin-Orbital Interaction
In this chapter, we study numerically the electronic transport properties and the
localization problems of a 2DEG with the Rashba spin-orbital interaction and disorder
potentials in the presence of a perpendicular magnetic ¯eld. The Hall conductance
is calculated numerically using a lattice model with the S-matrix, the Green function
methods and Landauer-BÄuttiker formulae. Transfer matrix methods are employed
in investigating the localization property of the system. We ¯nd that the quantized
charge Hall conductance is intact in the presence of the Rashba spin-orbital interaction
(SOI) and a weak disorder potential. In most cases, each Landau level gives a Hall
conductance plateau of value e2=h. However, if crossing of two Landau levels occurs
near the Fermi surface, due to the competition of Zeeman energy splitting and spin-
orbital interaction, the Hall conductance will jump by 2e2=h.
3.1 Introduction to Rashba Spin Orbital Interaction
Recently spintronics, the manipulation on the electrons' spin degrees of free-
dom, is believed to have a promising future in the information technology 19. Nowa-
days, Utilizing the spin orbital interaction (SOI) to manipulate the spin degrees of
freedoms of electrons has been advanced. Several spin devices have also been theoret-
ically designed. Experimentally, Rashba SOI which comes from the lack of structure
inversion symmetry 20 has been successfully controlled by the applied gate voltage
or with some speci¯c design of heterostructures 21;22. This interaction competes with
Zeeman splitting in perpendicular magnetic ¯eld was detected by the Shubnikov-de
Hass oscillation 21;23.
13
14
The Hamiltonian of Rashba SOI in the 2D x-y plane without magnetic ¯led
takes the form
HR = ®^z ¢ (¡!¾
£
¡!P
) = ®(¾xpy ¡ ¾ypx) (3.1)
where ® is the strength of Rashba SOI strength and ¾ is the Pauli matrix.
Shen et al. 24 found that Hall conductance is una®ected by the Rashba coupling
analytically with perturbation theory without disorder. Therefore, will Hall e®ect
remain intact in a 2DEG system with both Rashba coupling and disorder becomes
an interesting issue. They analytically calculated the Energy eigenvalues (LLs) of a
2DES with Rashba SOI, magnetic ¯eld and Zeeman splitting ¯eld. Their results are
given as:
²ns = ~!
Ã
n +
s
p
(1 ¡ g)2 + 8n´2
2
!
(3.2)
with s = §1 for n ¸ 1 and s = 1 for n = 0. ´ = ¸mlB=~2 is a e®ective Rashba SOI
strength. The relation of Ens and ´ is given in Fig. 3.1.
Figure 3.1. Landau Levels of an electron as functions of ´ for g = 0:1 is the Zeeman
splitting energy. The cross of the LLs are indicated with arrows in the
graph.
Figure 3.1 is taken from Ref. [24]. Without Rashba SOI, each LL is split into
two (spin up and spin down) LLS, because of Zeeman splitting. But with varying of
Rashba SOI and magnetic ¯eld, the LLs will cross each other. And they also applied
15
Kubo formulae to verify that Hall conductance is intact in the presence of Rashba
SOI.
But if we consider such a system with disorder potential, will the crossing of
LLs happen and will the Hall conductance remains the same as before become the
main topic for this chapter.
3.2 Tight binding model for a 2DES with Rashba SOI and a perpendicular
magnetic ¯eld
x
y
0
i
i-1
i+1
j-1 j j+1
Figure 3.2. The sketch for 2D lattice model with i stands for row indices and j for
column indices. i and j may also labeled as in y and x axis. The path
for an electron to jump one unit box is indicated.
First let's consider a simple 2DES with the Hamiltonian given by
H0 =
¡!p
2
2m
+ V (x; y) (3.3)
The tight binding approximation discrete lattice model for the above system can be
expressed as:
H0 =
X
ij¾
²ijay
ij¾aij¾ +
X
ij¾
t[(ay
i§1j¾aij¾ + h:c) + (ay
ij§1¾aij¾ + h:c)] (3.4)
with i and j is the label of rows and columns of the 2D lattice model as depicted in
Fig. [3.2]. So ²ij is the on site energy for lattice coordinate (i; j). We set ²ij = 0
16
as a reference energy and ²ij distributed uniformly in [¡w=2;w=2] for disordered on
site potential. ¾ =" or # is the lattice spin indices. If the system size is N £ N, the
total Hamiltonian matrix is 2 £ N £ N (2 comes from the di®erent spin). t = ~2
2m¤a2
is the nearest neighbor hopping element with the lattice space a. ay
i;j and ai;j are the
creation and annihilation operators for the electrons on the site (i; j). The matrix
representation of the Hamiltonian can be expressed as:
H(k; l) =
8>>>>><
>>>>>:
²kl; k = l
t; k; l being the nearest neighbor
0; otherwise
(3.5)
Next we will deduce the tight binding Hamiltonian for the Rashba SOI.We know
that the momentum operator in coordinate basis can be expressed as p = ¡i~d=dx.
For a local momentum operator acting on the local wave function can be expressed
as:
hxj j^pxj jªi = ¡i~
dª(x)
dx
jx=xj = ¡i~
ª(xj + a) ¡ ª(xj ¡ a)
2a
(3.6)
For the last equation, we just use ¯nite di®erence to approximate the derivative
operators and also assuming that lattice spacing a is really small. So local momentum
operator ^pxj reads:
^pxj = ¡
i~
2a
(jxjihxj + aj + jxjihxj ¡ aj) (3.7)
In the second quantization form, the above expression can be written as:
^pxj = ¡
i~
2a
(ay
jaj+1 ¡ ay
jaj¡1) (3.8)
For the local momentum operator for y, we just need to change j to i.
In the Rashba Hamiltonian, there is a term include ¾xpy. If we substitute local
momentum operator pyi into the above term, you ¯nd something like ¾xay
iai+1. Now
let's see how to write this in second quantization form.
¾xay
iai+1 = (ay
i;"; ay
i;#)
0
@0 1
1 0
1
A
0
@ai+1;"
ai+1;#
1
A = ay
i;"ai+1;# + ay
i;#ai+1;" (3.9)
17
Hence the Rashba Hamiltonian in a tight binding model gives:
HR = ~¸[(ay
ij"aij+1# ¡ ay
ij#aij+1") ¡ i(ay
ij#ai+1j" + ay
ij"ai+1j#) + H:c:] (3.10)
with ¸ = ®=2a is the strength of Rashba SOI. In this expression, we can see that
Rashba SOI will lead to the hopping of the nearest neighbors sites with di®erent
spin. 25;26
Including a perpendicular magnetic ¯eld for such a system, we only need to add
an additional phase factor to t and ¸. Such as t can be expressed as t¢exp[ie~
~A
¢(~rk¡~rl)],
~A
is the vector potential. When writing the program, we set up the hopping term
tkl as electron jumping from site l to site k. And choose the vector potential as
~A
= (¡By; 0; 0). So only the hopping term in x direction needs an additional phase
factor.
Let's look at an example, an electron hops counterclockwise from site (i; j) to
(i+1; j) to (i+1; j ¡1) to (i; j ¡1) and ¯nally back to (i; j) (indicated in Fig. 3.2).
The additional phase factor for electron hopping from site (i+1; j) to (i+1; j ¡1) is
(i + 1)eBa2=~ and hopping from site (i; j ¡ 1) to (i; j) the phase factor is ¡ieBa2=~.
So the total phase factor for a whole loop hopping is eBa2=~, which is the same as
e©=~ with © is the magnetic °ux through a unit box.
Also the on site potential energy is going to be changed under the magnetic ¯eld
due to Zeeman splitting energy. The on site spin up energy is changed to ²ij + ¢z=2
and spin down changed to ²ij ¡ ¢z=2 with Zeeman splitting energy ¢z.
3.3 Landauer-BÄuttiker Formulae for a Four Terminal Device
We study the transport properties for the 2DES with four terminals. The
model is given in Fig. 3.3. A square lattice is connected with four semi-in¯nite leads
to its four edges. Disorder potential, Rashba SOI, a perpendicular magnetic ¯eld and
Zeeman splitting ¯eld only exist in the central 2DES.
The current °owing out of the lead p is given by Landauer-BÄuttiker formulae
18:
Ip =
X
q
Gpq[Vp ¡ Vq] (3.11)
18
Here Vq is the voltage applied on the lead q. For the four terminal device, the relation
of current and voltage at each lead can be written as:
0
BBBBBB@
I1
I2
I3
I4
1
CCCCCCA
=
0
BBBBBB@
P
i=2;3;4 G1i ¡G12 ¡G13 ¡G14
¡G21
P
i=1;3;4 G2i ¡G23 ¡G24
¡G31 ¡G32
P
i=1;2;4 G3i ¡G34
¡G41 ¡G42 ¡G43
P
i=1;3;4 G4i
1
CCCCCCA
0
BBBBBB@
V1
V2
V3
V4
1
CCCCCCA
(3.12)
Figure 3.3. Sketch illustration of our numerical calculation model with four semi-in¯-
nite leads. The Rashba SOI, disorder potential and magnetic ¯eld only
exists in the central unshadowed region.
We calculate the voltage di®erence V2 ¡ V1 between leads 2 and 1 in the y
direction and let a unit charge current passing through the system in the x direction
with I3 = ¡I4 = I. The Hall conductance is de¯ned as
GH1 =
I
V2 ¡ V1
: (3.13)
We can also reverse the current and voltage terminals. Let the unit charge current
°ow in y direction with I1 = ¡I2 = I, and de¯ne Hall conductance as
GH2 =
I
V3 ¡ V4
: (3.14)
If disorder is not presented in system, it's easy to understand that GH1 = GH2 from
the system rotation symmetry. But in our model, the disorder potential in the system
19
will break rotation symmetry that makes GH1 not equal to GH2. The di®erence of
these two depends on the strength of °uctuation induced by the disorder. Therefore,
all my following numerical calculation about Hall conductance is the average of GH1
and GH2.In our numerical calculation, we also ¯nd that GH1(B) = ¡GH2(¡B) and
GH2(B) = ¡GH1(¡B), if we reverse the direction of magnetic ¯eld(only change the
sign of B and g in the program). This result is exactly the same as given by the
reciprocity relation27. Reverse the magnetic ¯eld and reverse the current and voltage
terminals, the measured Hall conductance is the same as before.
The above conductance matrix G is a singular matrix, because its determinant
is zero. Without loss of generality, we can set the voltage on one terminal to be zero
(V4 = 0). So the conductance matrix will become a 3 £ 3 matrix instead of 4 £ 4 in
the form: 0
BBB@
I1
I2
I3
1
CCCA
= G3£3
0
BBB@
V1
V2
V3
1
CCCA
(3.15)
This G matrix is not a singular matrix, so we can inverse it to resistance matrix
with R3£3 = G¡1
3£3 0
BBB@
V1
V2
V3
1
CCCA
= R3£3
0
BBB@ I1
I2
I3
1
CCCA
(3.16)
For example, consider a unit current °owing out of lead 3 into lead 4. I3 = 1.
In the voltage probe, no net current °ows in. I1 = I2 = 0.
0
BBB@
V1
V2
V3
1
CCCA
= R3£3
0
BBB@
0
0
1
1
CCCA
(3.17)
The Hall conductance can be written as:
¾xy =
I
V1 ¡ V2
=
1
R13 ¡ R23
(3.18)
We can numerically prove that the current °owing into lead 4 equals -1. Or we can
set lead 3 and 4 as voltage probe and let a unit current °owing out from lead 1 to 2.
20
I1 = ¡I2 = 1, I3 = 0. 0
BBB@
V1
V2
V3
1
CCCA
= R3£3
0
BBB@
1
¡1
0
1
CCCA
(3.19)
In this case the Hall conductance reads:
¾xy =
I
V3
=
1
R31 ¡ R32
(3.20)
Now, we see that if the G matrix is calculated, we can get the Hall conductance
for such a system. When the thermal energy kBT is much smaller than the Fermi
energy, with scattering matrix theory we can get 27 Gpq = 2e2
h Tpq . where Tpq is the
transmission probability between lead p and q. And it is determined by 28;29
Tpq = Tr[¡pGR¡qGA]: (3.21)
Here ¡p = I[§Rp
¡§Ap
], and GR(A) is the retarded(advanced) Green Function given by
GR(A) =
1
E ¡ H § i´
: (3.22)
§Rp
is the retarded self-energy of terminal p, which equals the conjugate of §Ap
. H
is the single particle Hamiltonian of the center square space minus the self-energy of
each lead. ´ is a very small positive number.
3.4 Self Energy Term for a Normal Semi-in¯nite Lead
Calculate the self term for a normal semi-in¯nite lead is a very trick part. So
in this chapter, we will explain this procedure in detail. For the details of Green
function technique, see Ref. [30].
First let's see what the self energy term is for a normal 2DES system with the
Hamiltonian:
H = H0 + HI =
X
i
²iay
iai +
X
hiji
(tijay
iaj + h:c) (3.23)
The ¯rst summation for lattice on site energy is de¯ned as unperturbed Hamiltonian
H0. And the second term is interaction Hamiltonian HI . The i, j summation in only
21
for nearest neighbor. The Dyson equation for the above system can expressed as:
Gr
ij = gr
ij +
X
kl
gr
ik§r
klGrl
j (3.24)
with g stands for the Green function for the unperturbed Hamiltonian. G is the Green
function for the whole system Hamiltonian including interaction. The retarded Green
function is Gr
ij(t; t0) = ¡iµ(t ¡ t0)hfai(t); ay
j(t0)gi. From the derivative of t on both
sides of the equation, we have
i
@
@t
Gr
ij(t; t0) = ±(t ¡ t0)hfai(t); ay
j(t0)gi ¡ iµ(t ¡ t0)hf[ai(t);H(t)]; ay
j(t0)gi (3.25)
And the commutation relation is
[ai;H] = ²iai +
X
k
tikak (3.26)
Substitute the above relation to Eq. 3.25, we have
µ
i
@
@t
¡ ²i
¶
Gr
ij(t; t0) = ±ij±(t ¡ t0) +
X
k
tik(t)Gr
kj(t; t0) (3.27)
(i
@
@t
¡ ²i)gr
ij(t; t0) = ±ij±(t ¡ t0) (3.28)
With a little trick, Eq. 3.28 becomes
Gr
ij(t; t0) = gr
ij(t; t0) +
X
k
Z
dt1 ¢
£
gr
ii(t; t1)tik(t1)Gr
kj(t1; t0)
¤
(3.29)
Compared this equation with Eq. 3.24, we get the self energy for such a system is
§r = tik (3.30)
Hence we see that if the self energy is only nonzero for the nearest neighbor hopping
element.
Now we consider about the self energy for a semi-in¯nite lead. Hamiltonian
H0 and HI are setup as depicted in the Fig. 3.4. The solid line connecting part is
H0 and the dashed line connecting part is HI . Because the lead is in¯nite long, the
Hamiltonian H0 and HI are both in¯nite dimensions. But we are only interested
in its N £ N part, which is only the ¯rst column, since only this layer is contacted
22
Figure 3.4. Illustration to calculate the self energy for a semi-in¯nite lead. The solid
line connecting system is considered as Hamiltonian H0, the dashed
line connecting part is interaction Hamiltonian HI .
with the 2DES. From Eq. 3.24, we rewrite the Dyson equation in the form Gr =
gr + gr§rgr + gr§rgr§rGr.
If we only consider the nonzero component of the self energy term, the above
equation becomes
Gr
ij = gr
ij +
X
k
gr
ik§r
k;k+NgrN
+k;j +
X
kp
gr
ik§r
k;k+Ngrk
+N;N+p§r
N+p;pGr
pj (3.31)
where i, j, k, p (from 1 to N) are only indices for the ¯rst layer. In the second term,
§r
k;k+N = t, which is just the hopping term. grN
+k;j = 0, since N + k and j belongs
to di®erent layer. In the third term, §r
k;k+N = t and §r
N+p;p = t. We also have
grk
+N;N+p = Gr
k;p, since the lead is in¯nite long. If we start from the second layer, the
unperturbed g is the same as perturbed G start from the ¯rst layer. Now the above
equation simpli¯ed to:
Gr
ij = gr
ij + t2
X
kp
gr
ikGr
kpGr
pj (3.32)
In a compact form, the above equation becomes
Gr = gr + t2gr(Gr)2 (3.33)
23
Or if we start the Dyson equation with Gr = gr + Gr§rgr, we will get Gr = gr +
t2(Gr)2gr. For a normal lead, we consider the lattice on site energy is zero, so the
unperturbed gr is expressed as
(gr)¡1 =
0
BBBBBB@
E + i´ ¡t 0 0
¡t
. . . . . . 0
0
. . . . . . ¡t
0 0 ¡t E + i´
1
CCCCCCA
(3.34)
This matrix can be diagonalized with a unitary transformation. The N £ N unitary
matrix U is:
U =
r
2
N + 1
0
BBBBBBBBB@
sin( ¼
N+1) ¢ ¢ ¢ sin( ¼k
N+1) ¢ ¢ ¢ sin( ¼N
N+1)
...
...
...
sin( ¼i
N+1) ¢ ¢ ¢ sin( ¼ki
N+1) ¢ ¢ ¢ sin( ¼Ni
N+1)
...
...
...
sin( ¼N
N+1) ¢ ¢ ¢ sin( ¼kN
N+1) ¢ ¢ ¢ sin( ¼N2
N+1)
1
CCCCCCCCCA
(3.35)
For the element in the matrix, Uik =
p
2=N + 1 sin [¼ki=(N + 1)]. So (gr)¡1 can be
diagonalized as:
(~gr)¡1 = Uy(gr)¡1U =
0
BBB@
E + i´ ¡ 2t cos( ¼
N+1) 0
. . .
0 E + i´ ¡ 2t cos( ¼N
N+1)
1
CCCA
(3.36)
Inverse the above equation, we got ~gr = UygrU and de¯ne ~G
r = UyGrU. The unitary
transformation for Eq. 3.32 becomes
~G
r = ~gr + t2~gr(~G
r)2 = ~gr + t2(~G
r)2~gr (3.37)
Compare the above equation, we have ~gr(~G
r)2 = (~G
r)2~gr. Since ~gr is a diagonal
matrix with ~gr
ii 6= ~gr
jj for di®erent i and j, we know that ~G
r is also a diagonal matrix.
Once we work out ~G
r, we can get Gr.
Choose the diagonal element in Eq. 3.36, we have
t2(~G
r
k)2 ¡ ²k ~G
r
k + 1 = 0 (3.38)
24
with ²k = (~grk
)¡1 = E + i´ ¡ 2t cos[ ¼
N+1k].
~G
r
k =
8><
>:
²k¡i
p
4t2¡²2
k
2t2 ; 4t2 > ²2
k
²k¡
p
²2
k¡4t2
2t2 ; 4t2 < ²2
k
(3.39)
Notice that the above two equations we only adopt the negative sign, since in the ¯rst
case, the imaginary part of the retarded Green function should be a negative number.
In the second case, at ²k goes to 1, the retarded function goes to zero. Finally, we
have the retarded Green function for a semi-in¯nite lead
Gr
ij =
X
k
Uik ~G
r
kUy
kj : (3.40)
From the reference 27, the self energy function is non zero only for the points
on the conductor that are adjacent to a lead. [Fig. 3.5] §r
p(i; j) = t2Gr
p(pi; pj) with
t is the nearest neighbor hopping term and Gr
p(pi; pj) is the retarded Green function
for the semi-in¯nite lead, which calculated before.
Figure 3.5. A conductor connected with a normal lead p on the left side. A point in
lead p is labeled Pi if it is adjacent to point i in the conductor. The
dashed line is the connection of lead with conductor. The hopping
phases in the junction are labeled. The total phase for a unit box in
the junction is zero. So there is zero magnetic ¯eld in the junction.
Now since magnetic ¯eld is inside the conductor, in the junction of lead with
conductor, the hopping term has to be changed (including an additional phase factor).
25
From section 3.2, we know that in the gauge A = (¡By; 0; 0), the hopping term t
in the x direction should include a phase factor. Suppose the hopping term in the
boundary of 2DES at x direction is ¡©, the hopping term at the junction should
include another phase factor to make sure that a unit box in the junction contains no
net magnetic °ux. Hence the magnetic ¯eld inside the lead is zero. We can choose
the junction hopping as t(pi; i) = t exp (i©). So the self energy function in the lead is
written as:
§r
p(i; j) = t¤(pi; i)Gr
p(pi; pj)t(pj ; j) = t2 exp [(j ¡ i)©]Gr
p(pi; pj) (3.41)
Also notice that in the above gauge, only hopping term in the x direction has an
additional phase, so the above self energy function is only needed for lead 1 and
2. Now equipped with all these information, it is easy for us to write a program to
calculate the Hall conductance for such a four terminal device. In the next section, we
will present our numerical results about transport behavior and localization properties
in this system.
3.5 Our numerical results about transport and localization properties
Figure 3.6. The normalized localization length L/M vs disorder strength for a long
bar of length 100000 and width M=12, 24, 40. The parameters are
setup as E=-2.0 and ¸ = 0:5.
26
Figure 3.7. The normalized localization length versus Fermi energy E with strength
of disorder w = 0:5, Zeeman splitting energy g = 0:6, B = e¼=4~ and
system width M=40 at di®erent Rashba strength.
In this section, we set t=1. So variables E, W, ¸ are in the unit of hopping t.
From Anderson scaling theory,31 we know that there is no metal insulator transition
(MIT) in the 2DES. Now if we include Rashba SOI, will MIT exist in such a system
becomes a very interesting question. In Fig. 3.6, we numerically calculated the
localization length L in a bar system of in¯nite length 100000 and ¯nite width M.
The method is a very established transfer matrix theory.32
In this graph, at weak disorder strength, L/M increases as M increases, indicat-
ing that with M goes to 1, L will diverge. So this corresponds to a extended metallic
phase. But when disorder strength is larger than the critical point (the cross of these
lines Wc = 6:3), L/M decrease as M increases. It means that with M goes to 1,
L will converge to some ¯nite value. So this is corresponds to a localized insulator
state. Hence we can see that MIT will exist for a 2DES system with Rashba SOI.
But if without Rashba SOI, there will be no cross of this lines, which means the
nonexistence of MIT in a ordinary 2DES. Similar results are already given in Ref.
[33].
27
The normalized localization length ¹» as a function of the Fermi energy is given
in Fig. 3.7. ¹» = »M=M, where »M is the ¯nite-size localization length in a sample
of size M £ M. In this ¯gure, the position of each peak on ¹» corresponds to the
extended states in a Landau level. The width of the peak is related to the Landua
level broadening due to the disorder. In the absence of the Rashba coupling, the
Zeeman energy splits the two degenerate Landau levels of spin-up and spin-down. In
that case, if the Zeeman splitting is smaller than the width of the peak, the two peaks
corresponding to spin-up and spin-down Landau levels are not distinguishable. As the
Rashba spin-orbit coupling increases, there is an energy crossing of the Landau levels
for certain values of the coupling constant ¸. If the Fermi energy is at the crossing
level of the extended states, the two peaks merge as one. In general, the Rashba
coupling competes with the Zeeman energy and reduces the Landau level splitting
due to the Zeeman term, which may induce the two peaks indistinguishable as their
energy separation becomes smaller than the width of each peak.
If the energy separation between the extended states in the two Landau levels
are so close in their energies that the distance of the two peaks is smaller than the
half width of the peak, we can no longer distinguish them in the graph. Hence they
are regarded as degenerated Landau levels.
The degeneracy of a lower LL of spin up and spin down appears at a higher
coupling ¸ compared to that of a higher LL. We can see in this ¯gure that the energy
crossing of the two peaks for the lowest Landau levels with spin up and down occurs
near ¸ = 0:4. While the energy crossing of the next Landau level occurs near ¸ = 0:2.
Note that there also exists energy crossings between di®erent Landau levels. At the
same time, we ¯nd that the energy of the spin-up peak of the lowest Landau level
(the ¯rst right peak in the upper most graph of ¯gure 2) remains the same in spite
of the changing of the spin-orbital interaction. These conclusions ¯t quite well with
the analytical results given by Shen et al.24
If disorder strength is larger than the critical Wc (in the insulator regime),
what will the LLs looks like. In Fig. 3.8, we ¯nd out that with the increasing of
disorder, the higher Landau level will strart to merge, due to the broadening of the
28
0 1 2 3 4
Fermi energy E
0
3
6
9
12
xM/M
w=2.0
w=3.0
w=5.0
w=6.5
Figure 3.8. Fermi energy versus normalized localization length at di®erent strength
of disorder with ¸=0.3, zeeman splitting energy g = 0:6 and system
width M=40.
Landau levels caused by the disorder. When w equals 6.5, the curve almost becomes
°at. So we can't distinguish any discrete Landau levels. The mixing of higher LLs
always happens earlier than lower LLs with the increasing of disorder strength. The
disappearance of IQHE will happens at strong disorder. Similar results are already
predicted by Sheng et al.33
In Fig. 3.8, we plot the Hall conductance versus Fermi energy (solid line) and
the normalized localization length versus Fermi energy (dotted line). We ¯nd that
in general each extended state contributes e2
h to the Hall conductance. However, the
degeneracy of two extended levels, if occurring at the Fermi level, gives rise to a
contribution 2e2
h to the Hall conductance. Since Hall conductance in the unit of 2e2=h
is the sum of all occupied degenerate LLs below Fermi energy. If the Fermi energy
lies in the energy gap between two consecutive Landau levels, the Hall conductance
is a well quantized integer. But if Fermi energy lies within a Landau level, the
Hall conductance exhibits an unusual large peak. We believed this kind of annoying
features is the result of the ¯nite-size e®ect of our numerical calculations.
29
2 2.5 3 3.5 4
0
5
l=0.2
G(e2/h)
xM/M
2 2.5 3 3.5 4
0
5
l=0.34
2 2.5 3 3.5 4
Fermi energy E
0
5
l=0.4
Figure 3.9. Hall conductance versus Fermi energy (dashed line) and normalized local-
ization length versus Fermi energy (solid line) with strength of disorder
w = 0:1 and Zeeman splitting energy g = 0:4. The size of the system
is 40 £ 40 and lead width is 20.
0.5 1 1.5
1
2
3
4
5
6
l=0.25
0.5 1 1.5
magtetic field B
1
2
3
4
5
6
l=0.3
Figure 3.10. Magnetic ¯eld versus Hall conductance with strength of disorder w = 0:1
and the Fermi energy E = 2:0. The zeeman splitting energy is
g = 1:2
¼ B. The size of the system is 40 £ 40 and lead width is 20.
30
The Hall conductance versus magnetic ¯eld is in Fig. 3.10. This is the situation
which is quite accessible to an experimental realization. With a decreasing of the
magnetic ¯eld, there is also a jump in the Hall conductance. (A decreasing of the
magnetic ¯eld leads to an increase of the ¯lling factor.) We ¯nd that there is a jump
of 2e2
h in Hall conductance at certain value of ¸. This ¸ corresponds to the point at
which the crossing of Landau levels happens at certain value of magnetic ¯eld.
In summary, we investigate the Hall conductance and localization problems in
a disordered 2DEG lattice model with Rashba spin-orbital interaction. We ¯nd that
the integer quantized Hall e®ect still exists at weak disorder. The degeneracy of
Landau levels as the result of Rashba SOI will contribute a 2e2
h plateau to the Hall
conductance. And MIT exist in a 2DES with Rashba SOI.
CHAPTER 4
Fractional Quantum Hall E®ect
Several interesting properties of fractional quantum Hall e®ect, such as Laughlin
states, fractional charge, quasi-particle statistics are reviewed in this chapter. The
quasi-particles excitation of fractional quantum Hall liquid has fractional elementary
charge. These quasi-particles are neither bosons nor Fermions, but anyons.
4.1 The Laughlin wave function for a º = 1
2n+1 state
In 1982, Tsui et al. ¯rst ¯nd that the Hall resistance exhibit the plateau at
h=3e2. (the plateau at magnetic ¯eld B=30T in Fig. 1.1) From chapter 2, we know
that at this case the lowest LL is only partially ¯lled. If we de¯ne the ¯lling factor
º =
N
BA=©0
(4.1)
N is the total number of electrons and BA=©0 is the number of °ux quanta. The Hall
conductance is thus ºe2=h. From Eq. 2.5, we know that if one LL is ¯lled, the ¯lling
factor º = 1. So in the above experimental results, º = 1=3, which means that LL is
one third ¯lled. To ¯nd the wave function for a º = 1=3 state, it's very useful for us
to ¯nd the many electrons wave function for º = 1 state.
Consider the Hamiltonian in Eq. 2.1, we change the vector potential to a
symmetry gauge (By=2;¡Bx=2; 0). The Hamiltonian thus becomes
H =
1
2m
[(px + eBy=2)2 + (py ¡ eBx=2)2] (4.2)
Now we de¯ne two operators as 34:
Q =
px + eBy=2
eB
; (4.3)
31
32
P = py ¡ eBx=2 (4.4)
It's easy for us to verify that [Q; P] = i~. The above Hamiltonian can be transformed
to
H =
P2
2m
+
1
2
m!2Q2 (4.5)
with cyclotron frequency ! = eB=m. Now we see that this equation looks like a
Harmonic oscillator. Thus we can de¯ne Ladder operator with:
a =
r
m!
2~
Q + i
r
1
2m!~
P (4.6)
ay =
r
m!
2~
Q ¡ i
r
1
2m!~
P (4.7)
with [a; ay] = 1. The Hamiltonian simpli¯ed as H = (aya + 1=2)~!. The eigenvalues
for such a system is
En = (n +
1
2
)~! (4.8)
This is just match the results in Eq. 2.4.
Now we change the x, y coordinate in a complex plane with z = x + iy and
¹z = x ¡ iy. We also have
@z =
1
2
(@x ¡ i@y) (4.9)
@¹z =
1
2
(@x + i@y) (4.10)
Hence
a = ¡i
r
2~
eB
(@¹z +
eB
4~
z) (4.11)
ay = i
r
2~
eB
(¡@z +
eB
4~
¹z) (4.12)
The ground state wave function in the complex plane can be evaluated from
the condition aj0i = 0
(@¹z +
eB
4~
z)ª0(z; ¹z) = 0 (4.13)
with solution
ª0(z; ¹z) = f(z) exp(¡
jzj2
4l2B
) (4.14)
33
with magnetic length l2B
= ~=eB and f(z) is an analytical function only depend on
z. We can choose a complete set of lowest LL wave function as:
ª0;n(z; ¹z) = zn exp(¡
jzj2
4l2B
) (4.15)
n start from 0 to in¯nity.
In a polar coordinates, Lz = ¡i~ @
@Á.
Lzª0;n(z; ¹z) = n~ª0;n(z; ¹z) (4.16)
n is actually angular momentum. The density of wave function actually is a ring shape
structure with a maximum at rn =
p
2nlB. And the ring width is ¢r = lB=
p
2n.
The area of this wave function is S = 2¼rn¢r = 2¼l2B
, independent of electron orbits,
which means that each electron occupy the same area.
Figure 4.1. The density of eigen wave function ª0;n(z; ¹z) is a ring shape.
Consider the case where N identical Fermions ¯lled N orbits (Pauli exclusive
principle). To ¯nd the total wave function for N identical electrons, we have to use
slater determinant to anti-symmetrize the many body wave function.
34
ª0(z1; ¢ ¢ ¢ ; zN) =
¯¯¯¯¯¯¯¯¯¯¯¯
1 ¢ ¢ ¢ 1
z1 ¢ ¢ ¢ zN
...
zN¡1
1 ¢ ¢ ¢ zN¡1
N
¯¯¯¯¯¯¯¯¯¯¯¯
exp
Ã
¡
PN
i=1 jzij2
4l2B
!
=
Y
1·i<j·N
(zi ¡ zj) exp
Ã
¡
PN
i=1 jzij2
4l2B
!
(4.17)
It has maximum angular momentum nmax = N ¡ 1. The density is uniform in such
model with n = N=2¼nmaxl2B
¼ 1=2¼l2B
. Filling factor º = n©0=B = 1.
Now, considering a case where magnetic ¯eld is large, so only the lowest LL
is partially ¯lled with º < 1. This state is actually highly degenerated state. The
degeneracy is N©!=N!(N ¡ N©)! (N is the number of electrons, N© is the number
of states in one LL). If we still try to solve FQHE in a non-interacting system, it's
impossible. Because if LL is partially ¯lled, electrons can be scattered into the empty
states. It's longitudinal conductance will be nonzero due to dissipation scattering.
Also without electron-electron interaction, the energy needed to add in an additional
electron is zero. This suggest that as the change of B ¯eld, the electron density will
change. So the Hall conductance will not exhibit a strict fractional plateau as the
change of B ¯eld. Therefore, the FQHE should be understood with electron-electron
interaction. And this interaction produce a gap between ground and excited state.
However, if the sample is too dirty, the FQHE will be destroyed. Thus we will discuss
the physics of FQHE of a partially ¯lled LL in a clean system.
Laughlin ¯rst guess a trial wave function for the model Hamiltonian
H =
X
i
"
(¡!p
i + e
¡!
Ai)2
2m
+ V (ri)
#
+
X
i<j
v(ri ¡ rj) (4.18)
The trial wave function for v = 1=2n + 1 (n is a integer) state is 35
ª2n+1(z1; ¢ ¢ ¢ ; zN) =
Y
1·i<j·N
(zi ¡ zj)2n+1 exp
µ
¡
P
jzij2
4l2B
¶
(4.19)
At n=0, the above equation returns to º = 1 case as given in Eq. 4.17. This wave
function matches well with the exact eigenstate of a number of electrons at di®erent
35
kind of interaction potential. Later, it was shown to be the exact ground state of
a class of Hamiltonians with nonlocal potentials.36 We also notice that this wave
function satisfy Fermion statistics, since 2n + 1 is an odd number. The total angular
momentum for the above wave function is
N(N ¡ 1)(2n + 1)
2
(4.20)
4.2 Properties of Laughlin Wave Function
The Laughlin wave function actually describes a uniform electron density
1
2¼(2n+1)l2B
with ¯lling factor º = 1
2n+1. To understand this property, we have to
apply plasmon analogy.
The density of such wave function can be written as:37
P(z1; ¢ ¢ ¢ ; zN) / jª(z1; ¢ ¢ ¢ ; zN)j2 = exp [¡¯V (z1; ¢ ¢ ¢ ; zN)] (4.21)
with ¯ = 2=m, m = 2n + 1 and
V (z1; ¢ ¢ ¢ ; zN) = ¡m2
X
i<j
ln jzi ¡ zj j +
m
4l2B X
jzij2 (4.22)
This is Boltzman distribution of N particles with charge m repelling each other with
2D Coulomb law, and attracted by a background negative charge with density ½Á =
¡1=2¼l2B
.
Now let's understand the 2D Gauss law (all ¯eld lines are con¯ned in a 2D
plane). In Gauss unit, it changes to
I
dl ¢ E = 2¼Q (4.23)
So E = Q=r, ª(r) = ¡Qln(r) and F = Q1Q2=r.
The potential energy of N particles with charge m in a 2D form is U1 =
¡m2P
i<j ln jzi ¡zj j, which is the ¯rst term. Then consider the negative background
charge act on charge m at the distance r. The net force is F = m(½Á¼r2)=r = m½Á¼r.
Their potential energy becomes U02
= ¡m½Á¼r2=2 = mr2=4l2B
. The total potential en-
ergy of charge m particle and the uniform background is U2 = m
P
jzij2=4l2B
(second
term).
36
From the perfect screening property of plasmon, local electrical density is neu-
tralized. So electrons should form a uniform density states to neutralize the back-
ground with electron density n satis¯es
mn + ½Á = 0; n =
1
2¼ml2B
(4.24)
Due to one component plasmon properties, numerical work have shown that at
low temperature with m a small odd number (m=3,5), the Laughlin state actually
describe the liquid state instead of crystal.38
4.3 Quasi-hole Excitation and Statistics
In this section, we explore the excitation of Laughlin liquid. Consider a in¯nite
thin magnetic solenoid pierce the sample at ». Then all electrons will try to stay away
from ». If the °ux is one quanta, all the electron angular momentum will increase by
1. Thus the wave function becomes
ª+
» (z1; ¢ ¢ ¢ ; zN) =
YN
i=1
(zi ¡ »)
Y
1·i<j·N
(zi ¡ zj)m exp
µ
¡
P
jzij2
4l2B
¶
(4.25)
This state actually describe a new state with a positive excitation at ».35 Numerical
results have shown that this excitation energy is around 0:026e2=lB.39
It's very interesting to notice that this quasi-hole excitation has a charge e/m.
From plasmon analogy, we ¯nd that
V»(z1; ¢ ¢ ¢ ; zN) = V (z1; ¢ ¢ ¢ ; zN) ¡ m
XN
i
ln jzi ¡ »j (4.26)
Now plasmon particles will feel the potential from an extra unit positive charge at
position ». Therefore, the plasmon particles with charge m at » will decrease by 1/m
to satisfy the neutralize condition, which means that electrons at » get smaller by
1/m. It's equivalent to a hole excitation with positive charge e/m.
The most interesting thing of these quasi-holes is that they are not Fermion
or Boson, but anyon. Arovas et al. ¯rst apply Berry phase technique to ¯nd the
statistics of these particles.40
37
The adiabatic theorem states that if the Hamiltonian is decided by a parameter
R(t), where R changes slowly with time, the particle will stay in the n eigenket of
H(R(t)) at time t, if it starts in the n eigenket of H(R(0)).
At time t, eigenket is given by
H(R(t))jn(R(t))i = En(R(t))jn(R(t))i (4.27)
State-ket satisfy
H(R(t))jª(t)i = i~
@
@t
jª(t)i (4.28)
When R(t) changes slowly, we expect jª(t)i proportional to jn(R(t))i.
jª(t)i = exp
·
¡
i
~
Z t
0
En(R(t0))dt0
¸
exp[i°n(t)]jn(R(t))i (4.29)
°n(t) is the Berry phase, de¯ned by:41
°n(t) = i
Z R(t)
R0
hn(R(t0))jrRn(R(t0))idR(t0) (4.30)
The path integration is the adiabatic process as the external parameter changes from
R0 to R(t). And °n is a real number.
Suppose the Hamiltonian include some potential pinning the quasi-hole. When
the pinning potential moves, the eigenstates evolve and Berry phase is accumulated.
Then we will be interested in the Berry phase acquired when a quasi hole move around
a close path. Thus »(t) changes adiabatically. The accumulated Berry phase is
° = i
I
d»2hª»j
d
d»
ª»i = i
X
i
I
d»2hª»j
1
» ¡ zi
jª»i (4.31)
If zi inside the integration loop,
H
d»2hª»j 1
»¡zi
jª»i = 2¼i, otherwise it's zero. So the
integration summation equals total number of electrons inside the loop times 2¼i.
Thus Berry phase becomes
° = ¡2¼An (4.32)
This phase should equals the phase accumulated as the quasi-hole wind around the
enclosed magnetic °ux ©. The phase becomes
q¤©
~
=
q¤BA
~
(4.33)
38
0 for Boson
p for Fermion
Others for anyon
Figure 4.2. Exchange statistics for Fermion, boson and anyon.
Then we have q¤ = nh=B = ºe, establish the previous quasi-hole charge.
Now consider a quasi-hole winding around another. The Berry phase becomes
° = ¡2¼An + 2¼A¢n (4.34)
The second one comes from the charge de¯cit at the second quasi-hole. A¢n =
q¤=q = º. So the extra phase (despite the ¯rst term of AB phase) acquired when
one quasi-hole winding around another is 2¼º = 2¼=m. The phase acquired when we
exchange two particles, equals half of phase when one particle winds around another.
So statistics angle of these two quasi-holes is ¼=m. Since m is an odd number bigger
than 3, these quasi-holes will have anyon statistics. The statistical angle for a boson
is 0, for fermion is ¼, others are anyon. However in 3D case, there will only be
fermion and boson, since the clockwise and counter-clockwise exchange of particles
are equivalent. Anyon only exist in 2D systems. Quasi-particle excitation in 2D
FQHE is the case that anyon appears in a physics system.
4.4 Outlook of Fractional Quantum Hall E®ect
In the later experiments, fractional plateaus at 2/3, 2/5, 2/7, 1/5...... are all
discovered. The common feature for these fractional plateaus is that they all have
the odd denominator. Haldane36 and Halperin42 proposed a hierarchical construction
39
for these states, such as 2/7 FQH state can be obtained by e/3 quasi-holes condensed
on the top of 1/3 state.
All these FQH states have di®erent charges and statistics, which suggested that
they belong to di®erent quantum phases. But all these states have the same symme-
try, which lead to the ine®ective of Landau symmetry broken theory to distinguish
them. In 1990, Wen ¯rst introduce the concept of topology order to describe this new
kind of order in FQH liquid.43 If the ground state degeneracy is robust against any
perturbation to the system, we can de¯ne the topology order from its degeneracy.
Recently, quantum computation from FQH states have been proposed.44 The
classical quantum computation is long thought to be impossible, since the wave func-
tion collapses during error checking. In 2003, Kitaev proposed that if a physical
system has topological degrees of freedom that are insensible to local perturbation,
then information contained there can be protected against perturbation from the
environment.45 Thus fault tolerant topological quantum computation is possible.
The quasi-particles at 1/3 states are Abelian anyons. since its Berry phase is
just a ordinary phase, it can not be applied for quantum computation. All we need
is non-Abelian anyon. In the non-Abelian case, the wave function has degeneracies,
so the Berry phase is actually a matrix. The non-Abelian state is currently the most
promising system to carry out topological quantum computation. Recent papers have
shown that 5/2 state might be a non-Abelian state 46 for quantum computation.
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VITA
ZHENYUE ZHU
Candidate for the Degree of
Master of Science
Thesis: QUANTUM HALL EFFECTS
Major Field: Physics
Biographical:
Personal Data: Born in Jingjiang, Jiangsu, P. R. China on November 13,
1981.
Education: Received the B.S. degree from Nanjing University, Nanjing,
China, 2002, in Physics. Completed the requirements for the degree of
Master of Science with a major in physics Oklahoma State University in
July, 2007.
Experience: Working as a research assistant in National Laboratory of
Solid State Micro-structures at Nanjing University from 2002 to 2003.
Employed as teaching or research assistant in Oklahoma State university
from 2003-2007.
Name: Zhenyue Zhu Date of Degree: July, 2007
Institution: Oklahoma State University Location: Stillwater, Oklahoma
Title of Study: QUANTUM HALL EFFECTS
Pages in Study: 42 Candidate for the Degree of Master of Science
Major Field: Physics
In this thesis, we ¯rst review the experimental observed quantum Hall e®ects in a
two dimensional electron system. The formation of integer quantum Hall e®ect can
be understood with the concept of ¯lling Landau Levels. When Fermi energy lies
between the gap of these Landau level extended states, the Hall conductance is a
quantized number.We also provide our numerical work about the Hall conductance in
such a two dimensional disordered lattice model with Rashba spin-orbital interaction.
Applying Landau-BÄuttiker formulae for a four terminal device, we ¯nd that the integer
quantized Hall e®ect still exists at weak disorder. The degeneracy of Landau levels as
the result of Rashba spin orbital interaction will contribute a 2e2
h plateau to the Hall
conductance. Metal-insulator transition exists in such a two dimensional system with
Rashba coupling. Finally, we introduce several basic exotic properties about fractional
quantum Hall e®ects.
ADVISOR'S APPROVAL: